Mathematics Seminar - Friday 24 June at 15:30

[Math Section]

M A T H E M A T I C S   S E M I N A R S  2016


Friday, 24 June, at 15.30 hrs.


Mark Watkins (University of Sydney)

Title
Class numbers of imaginary quadratic fields



Abstract:
The class number problem of Gauss inquires about the class number 
(denoted h) of an imaginary quadratic field, which is one measure of the 
failure of unique factorization. He found there were 9 fields with class 
number 1 (the last being Q(√-163), and conjectured there were no more. 
This case of h=1 was resolved by very different methods by Heegner and 
Baker (who won the Fields Medal) and Stark in the 60s. Soon after, 
Goldfeld proposed a plan to effectively show that h→∞, his idea being 
dependent on the existence of an elliptic curve having analytic rank 3 
(and meeting other minor technical conditions). The proof of this was 
then completed in the 80s by Gross and Zagier, involving a height 
formula for Heegner points. We discuss an improvement to Goldfeld's 
theorem, showing that an elliptic curve of analytic rank 2 suffices.



VENUE:  Luigi Stasi Seminar Room (ICTP Leonardo da Vinci Building, first 
floor)